I understand the concept of symmetry in geometry, but what do theoretical physicists mean when they speak of symmetry?
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It means that if you perform an experiment, that experiment should work exactly the same way if you mirror everything. In fact, that applies to time, too. The experiment should work the same way forwards and backwards in time. Now, doing things “backwards” in time requires you to do things like drive south instead of north. But as long as you’re careful, you shouldn’t be able to tell if a video of the experiment is running forward or backwards.
But here’s the thing. Sometimes, very rarely, it doesn’t work the same way backwards! That’s called “symmetry breaking” and tells us a little bit about why time flies in one direction and not the other.
Well that depends on the context because it can mean many things. A big issue with physics, though, is a specific kind of symmetry known as CPT Symmetry.
C stands for charge, P stands for parity, and T stands for time.
For C-symmetry, if you were to take every particle and swap it with its antiparticle (electrons for positrons, protons for anti-protons, etc.) then nothing changes.
For P-symmetry, if you were to physically mirror the universe along any of the three spatial axes, then nothing changes.
For T-symmetry, if you were to reverse the direction of time, then nothing changes.
You can consider these symmetries individually, or in combination with each other.
For example, some system might not be C-symmetric or P-symmetric but might be CP-symmetric. That is, if you *just* swap every particle for its antiparticle *or* mirror it along some spatial axis, then it is different, but if you do both, then nothing changes. This would be called CP-symmetry.
Currently, the universe, on the whole, isn’t C, P, or T symmetric. And it isn’t symmetric for any combination of two of them. But it is believed to be symmetric for all three, known as CPT-symmetry. So if you were to replace every particle in the universe for its anti-particle *and* mirror it about a spatial axis *and* reverse the direction of time, then it would appear the same as this universe.
A symmetry describes any kind of transformation that maps back to the original state. If you have a mirror symmetry, then performing that mirroring doesn’t change the image. If you have a rotational symmetry then rotating by the corresponding angle doesn’t change the image.
In physics this works similar: Spatial translation symmetry means that no matter where you are, the rules of the universe are the same. Spatial rotation symmetry means that no matter which direction you look, the rules of the universe are the same. Time translation symmetry means that no matter how early or late you look, the rules of the universe are the same.
There are also some symmetries that aren’t true for our universe. Parity symmetry means that if you mirror space in one dimension, the rules of the universe stay the same – but the weak interaction isn’t parity symmetric. Charge symmetry means that if you replace every particle with its antiparticle, the rules of the universe stay the same – but that’s not the case either.
3 symmetrys:
1. Charge symmetry: We have matter and anti-matter. Our universe is made of what we called “matter” and not antimatter. But the universe should work just fine if everything was anti-matter instead.
2. Parity symmetry: This is about mirroring the 3 physical axises.
3. Time symmetry. Basically it should be possible to have a universe where time went the opposite way.
Each of these symmetries have actually been broken. A “fix” to this is supersymmetry that says that all three symmetries cannot be broken together.
I am seeing a lot of CPT Symmetry discussion here. And that is a great example but I figured I would hop in with another.
I want to discuss U(1)XSU(2)XSU(3) symmetries, because, in my opinion, that is where symmetries in physics really shine. In physics symmetries mean something very similar to in geometry, it is all about how changing one aspect of a system (or a shape in geometry) doesn’t change the overall system. A good example in geometry is a circle, which is rotationally invariant, it has rotational symmetry. You can spin a circle by any amount and it will still look the same.
In quantum physics fundamental particles are described as a waves. With a typical wave it will have ‘global phase shift invariance’ which is a fancy way of saying that if you shift a wave from left to right, or right to left, without changing the amplitude (height) of the wave or the wavelength (distance between peaks) then the particle being described wont change. Just like with the circle we have a symmetry, you can shift the whole wave back and forth all you want and it wont change the particle.
That much is pretty normal, but the fun idea started when they worked out what it would mean for a wave to have ‘*local* phase invariance’ this is where you shift each point on the wave to the left or right by *different amounts* this will completely destroy the wave, it will be all disconnected and a total mess.
But that is where things get really cool, I have no idea who figured this out or how, but they realized that you can add a term to the equation for your wave function that will offset the issues created by any local phase shift. This equation then describes a particle that is invariant to local phase shifts. And it turns out the term you need to add describes how an electrically charged particle self interacts via the electromagnetic force, all electrically charged particles will have this self interaction term, and that shows that *what it means for a particle to be electrically charged is that it has local phase invariance.*
Fundamental forces come from symmetries. What I described above is a simplified version of U(1) symmetry and it is reasonable for the electromagnetic force. SU(2) symmetry is another symmetry that is responsible for the weak force and SU(3) is the one for the strong force. In quantum physics fundamental forces are completely tied to and rise from symmetries.
There are lots of viewpoints one could take. One is that they represent invariances: it doesn’t matter when I do an experiment – I should always get the same result – because of time translational invariance. This is time translational symmetry. It doesn’t matter where I do an experiment because of space translational invariance – this is spatial symmetry.
The reason physicists like symmetry so much is that they represent conservation laws. Time translational symmetry implies conservation of energy. Space translational invariance implies conservation of momentum. And the belief is that all physical conservation laws have a corresponding symmetry.
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